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\author{Francisco Aspillaga \and Alejandro F. Mac Cawley \and Sergio Maturana}
\title{A Bulb Procurement Planning Model for a Chilean Lily Flower Producer}
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\begin{document}
\maketitle
\begin{abstract}
The procurement of raw materials in the flower industry involves a large number of technical, biological, and economic variables along with a considerable level of capital. For this reason in the present paper we develop a Bulb Procurement Planning (BPP) mixed integer optimization model to support the procurement planning process. The model incorporates aspects such as substitution, precocity, risk diversification, seasonality, and inventory levels, among others. We implemented the model using the production planning data from a real firm. The results of the BPP model and the executed plan were compared with respect to costs, container programming, and  inventory levels.

\smallskip
\noindent \textbf{Keywords.} Aggregate production planning; mixed integer linear programming; bulb acquisition\
\end{abstract}

\section{Introduction}

The world floriculture  market has significantly grown in the past 15 years, from USD 8.5 billion in 2001 to USD 20.6 billions in 2013 \citep{Rabo_2015}. Cut flowers has been the main group within the global floriculture trade, where the exchange of products between countries has become a common practice. As part of this exchange, the countries located in the southern hemisphere, present a competitive advantage of having counter season production with their main markets, such as the United States, Europe, and Japan. This advantage enables the countries located in this hemisphere to get to the consumer with a high quality product in a period when international prices are higher. 

Chile still has a small share of the international cut flower market with a 0.03\%, being the US the main market with 82.8\% of the exports \citep{Odepa_2014}. Nevertheless, Chile presents a number of other advantages for the flower production, where we can mention the agro climatic characteristics, which vary enormously along the long extension of the country (From parallel 24 to 55) allowing a large number of species to be grown. The Mediterranean weather, with cold winters and moderate temperatures during the summer season, provides excellent conditions for the development of bulb species. Finally, having the Andes Mountains to the east, the Atacama dessert to the north and the Pacific ocean to the west; provides a strong  fitosanitary barrier that helps to keep out pests. These factors allow Chile to export to markets with high quality standards. Chile is the fourth exporting country in the world of flower bulbs, with a 1.8\% share of the international bulb flower market and its considered to be an attractive country for its  production \citep{Odepa_m_2014}

Because of the long distance between Chile and its main export markets of flowers, USA and Netherlands, and also to the import markets of bulbs, which is Netherlands, transportation costs strongly impact the production costs of the flowers. Since most of the best quality genetic material is imported, with high costs per unit, the bulbs generally comprise about 48\% of gross margins \citep{Kang01102013}. The high bulb purchase and transportation costs  make it necessary to have a bulb procurement planning  support system.

The purpose of this work is to develop a bulb procurement planning mathematical model of a Chilean lily flower producer, which will allow the development of a procurement planning support system.  To accomplish this objective, we formulated a mixed integer programming model, where the objective function was to minimize the relevant costs incurred in the procurement of bulbs subject to a number of constraints. The model was solved with data obtained from a Chilean producer and the results were compared with the actual costs of the plan carried out by this company in order to determine the benefits that might have been obtained had the model been used to plan the procurement of bulbs.

The contribution of this paper is to provide insight into the bulb procurement planning process required by a flower producer; develop a mathematical model to support the bulb procurement planning process, and test it. This model could be the basis of a computer system that could  help lily flower producers develop better procurement plans, more quickly, and with less effort.

\section{Bibliographic Discussion}

Mathematical models have been proposed for many years to support agricultural planning. According to \citet{dent1986}, optimization models should incorporate the following three elements:  (1) an objective function that minimizes or maximizes a function; (2) a description of the activities of the system, through the coefficients that represents the productive responses of the system; and (3) a set of constraints that define the operational conditions and limits of the productive system.

Later on \citet{glen1987} provides an extensive review of mathematical models applied to agricultural planning. There are also applications of production planning in the forest sector, described by \citet{weintraub1996},  \citet{laroze2000}, and \citet{palma2001}. Finally, \citet{broekmeulen2002} and \citet{caixeta2002} present different mathematical models to support decisions in the flower industry. The first compares three different optimization algorithms for the treatment of bulbs in order to minimize the cost of batch movement during treatment. The second describes an implementation of a production planning system, based on a a lineal programming model whose objective is to maximize the total contribution margin of a flower production company. For an extensive review on the application of planning models in the agri-food supply chain see \citep{ahumada2009}


\cite{Shukla2013}

Also cite \cite{ferrer2008}. 

\section{Bulb Procurement Planning}

In the present paper, we propose a Bulb Procurement Planning model (BPP), which finds the best combination of resources necessary to fulfill the demand of flowers with the minimum total cost. The model allows the purchase manager to find the combination of varieties that minimizes total cost. Bulbs are purchased from international suppliers in different periods during the year. It also optimizes the shipment organization, the quantity and type of bulbs to be purchased, programming the quantity of inventory to be held by the company.

The total procurement cost of the bulbs includes cost of the bulbs, the transport costs and the inventory costs of the bulbs. The planning horizon for the model is one year, since the purchasing negotiation with the bulb suppliers is carried out once a year. 

To better understand the procurement problem faced by the purchase manager, we will describe in more detail the procurement process.  Figure \ref{structure} shows the main stages of the current bulb procurement process. In stage I the feasibility of substituting some of the bulbs with lower cost ones, is analyzed. This feasibility depends on the characteristics of the flower market and of the bulbs. The purchasing cost of the different types of bulbs is also analyzed as well as the risk diversification, which is the percentage of the bulbs that are supplied by only one supplier. The idea is that if this percentage is too high, there is a higher risk associated to buying defective bulbs from that supplier. Therefore a good plan should require different suppliers for each bulb variety in order to reduce the risk. How much is safe to buy from each supplier depends on the their characteristics. After analyzing all these elements, a preliminary plan is constructed to meet the flower demand that the company expects during the planning horizon. Of course, having a good demand forecast is critical for the procurement plan. 

\begin{figure}[tbp]
\includegraphics[width=6.0in]{Fig1.pdf} 
% \includegraphics[width=3.9413in,height=2.3in]
\caption{Structure of the Bulb Procurement Process }
\label{structure}
\end{figure}

In stage II, the plan is checked for availability of the bulbs that need to be purchased. If the required bulbs are available, we proceed to stage III. Otherwise we need to substitute some of the bulbs with other ones, if it is possible, or else we cannot fulfill all the demand. In stage III the plan is analyzed to check the precocity and season availability of the
specified varieties. If the precocity or seasonal availability is not met, it is necessary to find a new substitute for the variety. In stage IV, the bulb shipment is organized. Currently the planner tries to minimize the number of shipments as long as the bulbs arrive on time to be planted. Finally in stage V, the plan is presented to the CEO in order to be approved and implemented.


\section{Mathematical formulation of the BPP model}

The BPP model is defined as the minimization of the purchasing, transportation, and inventory costs subject to different types of constraints. There are technical constraints, such as the maximum substitution of varieties, availability, production precocity of the varieties, risk diversification, and season availability of the bulbs. The model also includes constraints to insure that the flower demand is met and that the bulbs arrive on time to allow them to be planted and to develop into flowers when needed to meet the demand. There are also constraints that allow a a maximum level of substitution of individual varieties and the availability of them. Finally, there are some constraints that enforce the producer's policy of reducing the risk of purchasing defective bulbs. This policy specifies a maximum percentage of the total bulbs that can be purchased from certain suppliers and a minimum percentage that has to be purchased from certain certified suppliers.

The total bulb procurement costs is the following:

\begin{itemize}
\item Purchase Costs: Bulbs purchase; at values FOB or Exwork, depending on the commercial agreement between supplier and buyer.
\item Transportation Costs: The freight, insurance and transport costs between port and production facility. Warehouse costs, if needed, are also included.
\item Inventory Costs: correspond to the immobilized capital due to the purchase of the bulbs.
\end{itemize}

The  indices of the model are the following: 
\begin{itemize}
\item $i$ for the set of varieties that are demanded, 
\item $p$ for the set of varieties that are purchased, 
\item $j$ for the set of calibers, 
\item $l$ for the set of buds or flowers,
\marginpar{{\footnotesize What is the difference between bud and flower?}} 
\item $k$ for the set of suppliers, 
\item $t$ for the time period, and 
\item $b$ for the geographical origin of the bulbs, e.g., southern Chile or Netherlands.
\end{itemize}

The parameters or data of the model are the following:

\begin{itemize}
\item $V_{pjk}$ is the cost of the variety purchased $p$, of caliber $j$, from supplier $k$,
\item $T_{b}$ is the cost of transportation cost from origin $b$ (by land or sea),
\item $Q_{pk}$ is the minimum quantity of bulbs of the variety $p$ assigned to supplier $k$ as a percentage of the total demand,
\item $C_{b}$ is the maximum number of trays that a container can carry from origin $b$,
\item $O_{kb}$ is the assignment of supplier $k$ to origin $b$ (1 if assigned, 0 otherwise),
\item $B_{pj}$ is the maximum quantity of bulbs of variety $p$ and caliber $j$ that a tray may contain,
\item $A_{pjk}$ is the availability of variety $p$, of caliber $j$ from supplier $k$,
\item $S_{pi}$ is the substitution feasibility of demanded variety $i$ by variety $p$ (1 if feasible, 0 otherwise),
\item $D_{itl}$ is the demand of variety $i$, in time period $t$, and bud $l$,
\item $G_{ijl}$ is the conversion factor of bud $l$ and caliber $j$, of the demanded variety $i$,
\item $M_{i}$ is the maximum convertibility of the demanded variety $i,_{.}$
\marginpar{{\footnotesize What is convertibility?}}
\item $E_{tb}$ is the seasonal availability in time $t$ from the origin $b$ (1 if available, 0 otherwise),
\marginpar{{\footnotesize Is this correct? E is binary}}
\item $I_{pj}^{I}$ is the initial inventory of variety $p$ of caliber $j$,
\item $R$ is the interest rate, and
\item $P_{itt^{\prime }}$ is the precocity indicator, which indicates that the bulb of demanded variety $i$ purchased in time period $t$ could be harvested in time period $t^{\prime }$.
\end{itemize}

The decision variables of the model are the following:
 
\begin{itemize}
\item $x_{pjktb}$ is the number of bulbs of variety $p$ and caliber $j$, to be purchased from supplier $k$, in time period $t$, from origin $b$. 
\item $z_{tb}$ is the number of batches to be transported from origin $b$ at time $t$.
\marginpar{{\footnotesize Batches of what? How many whatevers in a batch?}}
\item $s_{pijtt^{\prime }}$ is the number of bulbs of demanded variety $i$ to be
substituted by variety $p$, of caliber $j$, to be purchased in time period $t$ and harvested in time period $t_{.}^{\prime }$. 
\item $n_{ijtt^{\prime }}$ is the number of bulbs of demanded variety $i$ not substituted, of caliber $j$, to be purchased in time period $t$ and harvested in time period $t^{\prime }$.
\item $I_{pjt}$ is the inventory of variety $p$ and caliber $j$, at the end of time period $t$. 
\item $y_{pjtt^{\prime }}$ is the number of bulbs of variety $p$, caliber $j$, planted in time period $t$, to be harvested in time period $t^{\prime }$. 
\end{itemize}%


The precocity parameter $P_{itt^{\prime}}$ allows us to define $F_{i}$ as the set of feasible combinations of time periods $(t,t^{\prime})$, such that: $F_{i}=\left\{ (t,t^{\prime })\mid P_{itt^{\prime}}=1\right\} $. Only the tuples $(t,t^{\prime })$ that are
defined in $F_{i}$ are used in the model, which greatly reduces the
dimensionality of the problem. We also define the sets $F_{i}^{P}$ and $%
F_{i}^{H}$ as the set of time periods for which there is a feasible
combination of time periods, that is, 
$F_{i}^{P}=\left\{ t\mid P_{itt^{\prime}}=1\text{ for some }t^{\prime} \right\}$. $F_{i}^{H}=\left\{t^{\prime }\mid P_{itt^{\prime}}=1\text{ for some }t\right\} .$

The objective function is the following:

\[
\mathit{min}\sum\limits_{p}\sum\limits_{j}\sum\limits_{k}V_{pjk}\sum%
\limits_{t\in F_{i}^{P}}\sum\limits_{b}\/\,x_{pjktb}+\sum\limits_{t\in
F_{i}^{P}}\sum\limits_{b}\/\,T_{b} z_{tb}+R\sum\limits_{p}\sum\limits_{j}%
\sum\limits_{k}V_{pjk}\sum\limits_{t}\/I_{pjt} 
\]

The objective function minimizes the total procurement cost.
The first part is the purchase cost of the bulbs, the second part is the transportation cost, and the third part is the cost of keeping inventory, which is the interest rate given by local banks to the company times the total bulb inventory. Note that $x_{pjktb}$ and $I_{pjt}$ are continuous variables while $z_{tb}$ are integer variables.

The following constraint enforces the fulfillment of the market demand in every time period:

\begin{equation}
\sum\limits_{j}\sum\limits_{p}\/\,G_{ijl}\sum\limits_{(t,t^{\prime })\in F_{i}}\/\left( s_{pijtt^{\prime }}+n_{ijtt^{\prime }}\right) \geq D_{itl}\;\;\forall \;t\in F_{i}^{H},i,l  \label{demandfulfillment}
\end{equation}

Bulbs can be partially substituted by another variety, taking into account the color of the flower in the same specie. The following constraints limit the total percentage of substitution in a given specie (\ref{substitutionconstraint}) and in the total demanded flowers (\ref{DemandFlowerConstr}):

\begin{equation}
\sum\limits_{j}G_{ijl}\sum\limits_{(t,t^{\prime })\in F_{i}}\/s_{pijtt^{\prime }}\leq \;\sum\limits_{t^{\prime }\in F_{i}^{H}}S_{pi} D_{it^{\prime }l}\;\;\forall \;p, i, l
\label{substitutionconstraint}
\end{equation}

\begin{equation}
\sum\limits_{j}G_{ijl}\sum\limits_{p}\sum\limits_{(t,t^{\prime })\in F_{i}}\/s_{pijtt^{\prime }}\leq M_{i} D_{it^{\prime }l}\;\;\forall \;t^{\prime }\in F_{i}^{H},il  \label{DemandFlowerConstr}
\end{equation}

The following expression computes the number of bulbs that can be harvested in time period $t^{\prime }$ that were purchased in time period $t$. 

\begin{equation}
y_{pjtt^{\prime }}=\sum\limits_{i}\/s_{pijtt^{\prime }}+n_{ijtt^{\prime}}\;\;\forall \;p, j\;(t,t^{\prime })\in F_{i}
\end{equation}

The following two constraints compute the inventory of bulbs at the end of each period during the planning horizon. Remember that $I_{pj\;\;}^{I}$ is the initial level of inventory.

\begin{equation}
\sum\limits_{k}\sum\limits_{b}\/x_{pjktb}+I_{pjt-1}=I_{pjt}+\sum\limits_{t_{F}}\/y_{pjtt_{F}^{\prime }}\;\;\forall \; p, j, t\geq 2
\end{equation}

\begin{equation}
\sum\limits_{k}\sum\limits_{b}\/x_{pjk1b} + I_{pj\;\;}^{I}  = I_{pj1}  \;\;\forall \; p, j
\end{equation}

The following constraint enforces that bulbs cannot be planted in nonfeasible time period tuples:

\begin{equation}
y_{ptt^{\prime }j}=0\;\;\forall \; p, j\;(t,\,t^{\prime })\notin F_{i}
\end{equation}

The following constraint enforces the availability of bulbs that each supplier $k$ has, for each variety $p$ and caliber $j$:

\begin{equation}
\sum\limits_{t\in F_{i}^{P}}\/\sum\limits_{b}\/\,x_{pjktb}\leq A_{pjk} \;\; \forall \;p, j, k
\end{equation}

The following constraint insures that each supplier $k$ has at least a certain percentage of the total demand, $Q_{pk}$ for each variety $p$. This diversifies the risk of having a supplier with quality problems.

\begin{equation}
\sum\limits_{j}\/\,\sum\limits_{b}\/\,x_{pjktb}\geq Q_{pk}\sum\limits_{j}\/\sum\limits_{h}\/\sum\limits_{b}\/x_{pjhtb}\;\; \forall \;p, k, t
\end{equation}

The following constraint relates the bulbs purchased with the containers that are shipped from each origin:

\begin{equation}
\sum\limits_{p}\/\sum\limits_{j}\/\sum\limits_{k}\/\/\,B_{pj} x_{pjktb} \leq \sum\limits_{k}\/\,C_{b}z_{tb}\;\;\forall \;b, t\in F_{i}^{P}
\end{equation}
\marginpar{{\footnotesize Need to explain better}}


Each origin $b$ has a seasonality, which means that some bulbs cannot be purchased at some origins in certain periods. This is indicated by the parameter $E_{tb}$, which is 1 if the bulbs are available and 0 otherwise. Note that if the bulbs are available the constraint is not active. 

\begin{equation}
\sum\limits_{p}\sum\limits_{j}\sum\limits_{k}\/x_{pjktb}\leq E_{tb}\sum\limits_{i}\sum\limits_{t^{\prime }\in F_{i}^{H}}\sum\limits_{l}\/D_{_{it^{\prime }l}}\;\;\forall \;t ,b
\label{SupplierAvailability}
\end{equation}

The following constraint is needed to enforce that it is not possible to purchase bulbs from a supplier $k$ at the origin $b$ if that supplier does not belong to that origin, which is specified by the parameter $O_{kb}$:

\begin{equation}
\sum\limits_{p}\sum\limits_{j}\sum\limits_{t\in F_{i}^{P}}\/x_{pjktb}\leq O_{kb}\sum\limits_{i}\sum\limits_{t^{\prime }\in F_{i}^{H}}\sum\limits_{l}\/D_{_{it^{\prime }l}}\;\;\forall \;k, b
\end{equation}

Finally, we have the following nonnegativity constraints:

\begin{equation}
x_{ijkt}\geq 0;\;\;z_{tb}\geq 0\in \mathbb{Z}^{+};s_{pijtt^{\prime }}\geq 0;n_{pjtt^{\prime }}\geq 0;I_{pjt}\geq 0;y_{pjtt^{\prime }}\geq 0\;
\end{equation}

\section{Results}

In this section we present how the BPP model can be used to aid decision making at the tactical level of a Chilean flower producing company. The Bulb Acquisition Planning model (BPP) was formulated in the algebraic modeling language AMPL, which is described by \citet{fourer1993}. The model considered: 54 varieties of Lilium spp., four bulb calibers, four blooming classifications, five suppliers, 78 weeks for the arrival of the bulbs,
52 weeks to flower time (demand) and two places of origin of the plant material (the Netherlands and Chile). 
\marginpar{{\footnotesize 78 weeks for the arrival seems excessive. Also explain what is flower time.}}
This resulted in a problem with 100,581 variables, of which 156 are integer, and 44,669 constraints, all linear. The model was solved using distributed NEOS optimization server version 4.0 (Czyzyk, 1998) and mathematical programming software XPRESS-MP (Dash Associates, 1999). The run-time for solving the problem was between 1.5 to 2.5 hours. 

We used the real bulb prices and the diversification and substitution policies used by the CEO at that time. The demand for each type of flower is provided by an expert advisor for the company and its for a period of one calendar year. The planning horizon must be quite long since the purchase decisions are made months in advance of cutting the flowers to meet the expected demand. We also need to consider the time required for ocean freight, the time required for planting, the precocity the different varieties, among processes that are needed to obtain a final cut flower. The production department provided the precocity of the varieties and the blooming calibers data.  

The results obtained using the BPP model are compared to the actual production plan used by the company in Table \ref{CostComparison}. This table shows that if the BPP model had been used, the total cost of operation would have been 2.15\% lower than those obtained using the actual plan. The difference between the bulb costs and the transportation costs of both plans are relatively small. However there is an important difference in the inventory costs, where the BPP model performs much better than the actual plan the company used. 

\begin{table}[tbp]
\caption{Comparison of Costs between BPP Model and the Actual Production Plan}
\label{CostComparison}%
\begin{tabular}{ccc}
\hline
& Actual Plan & BPP Model \\ \hline
Total Costs & 100\% & 97.85\% \\ 
Bulb Costs & 100\% & 99.59\% \\ 
Transportation Costs & 100\% & 103.33\% \\ 
Inventory Costs & 100\% & 36.34\% \\ \hline
\end{tabular}%
\end{table}

In Figure \ref{WeeklySchedule} we show the actual shipments during the entire planning horizon, compared to those that would have resulted if the BPP model had been used.
\marginpar{{\footnotesize Why 40 weeks instead of 52?}}
Note that decisions are being made in period 0. However the figure shows the previous 20 weeks since there are many shipments that have not yet arrived, but will within the planning horizon. Therefore it is necessary to take them into account. The lower part of the figure shows when the shipments arrive, and the number of containers in each shipment. The upper part of the figure shows the inventory level, in number of bulbs, at each time period. The figure clearly shows that the BPP plan has a lower inventory than the actual plan during the entire planning horizon. 

\begin{figure}[tbp]
\includegraphics[width=5.0in]{Fig2cp.png}
\caption{Comparison of the weekly container schedule and the number of bulbs in inventory obtained from the BPP model with the executed plan. Numbers below each container indicate the number of containers.}
\label{WeeklySchedule}
\end{figure}


The BPP model also specifies which substitutions have to be made, how many bulbs to plant, and when, so demand is met. Finally, it also specifies how much has to be planted, assigning different varieties for planting at specific times to meet the demand.


\section{Discussion}
 

Before the BPP model was implemented, decision makers divided the planning process into the stages that were shown in Figure \ref{structure}. First they decided which bulbs to purchase, based on the cost of the different types of bulbs quoted by the suppliers, and they took into account possible substitutions and diversifying their suppliers to reduce the perceived risk. In the second stage, the decision makers checked whether the varieties that were being purchased would be ready on time to meet the demand, and also if the seasonalities, and availabilities of the suppliers would enable them to supply all their purchases. Finally, they put together the shipments, in order to implement the decisions made in the first two stages, and to make sure that the demand would be met. The BPP model is able to take into account al these  aspects and find an optimal solution, which would be very difficult for a decision maker without some analytic support.

When analyzing the results of using the manual method versus the BPP model
shown in Table \ref{CostComparison}, it is clear that there is very little
difference between the two, with respect to the costs of the bulbs. A
possible explanation is that since it is the first element that decision
makers take into account, they very much optimize this part of the
production plan. However, when the inventory cost is analyzed, the
difference is very large in favor of the plan developed using the BPP model,
which generated savings of 63.66\% with respect to the manual method. This
reduction was possible thanks to a better shipment planning, although it
also meant a slight increase in transportation costs, of 3.33\%. 

About 85\% of the savings that could be generated by the BPP model are due
to a reduction in inventory levels of the bulbs. Since bulbs are relatively
expensive, reducing the capital invested in inventory generates important
financial savings. 

The plan generated by the BPP model was discussed with the executives of the
participating company, who checked that it met all the relevant criteria for
substitutability, supplier diversification, and met the demand. The
executives agreed that the plan met all their requirements. 

In summary, the development of the BPP model made three contributions to the
participating company. It helped the decision makers to better understand
the main issues that should be taken into account when they develop a
production plan, and how they relate with each other. The model also helped
integrate all the elements necessary to make an optimal decision, and it
explicitly shows the different production costs, including the inventory
cost. Finally, the model can be used as an operational tool that can help
the participating company improve their production planning process, thus
reducing their costs. 

It is important to note that the BPP model is deterministic, and assumes
that all the parameters are known with certainty. This is an assumption that
is not very realistic with respect to the demand, so it would be interesting
to develop in the future a model that could better handle the uncertainty in
the demand. For example, a robust model, such as the one described in \cite{Bohle2010} could be considered for this problem, since there is very little knowledge about distribution probabilities of the uncertain parameters. 

\section{Conclusion}
%% Comparar con el paper en espagnol

Production planning in the flower industry is complex, since many different
elements must be taken into account. There is the possibility of
substituting one variety for the other, it is also possible to buy bulbs
that can be harvested sooner, there are different origins, and suppliers
with different characteristics. It also takes into account different
precocities, risk diversification, and seasonal availabilities. The current
method used by companies in Chile to plan their production process, breaks
up the problems in stages which are dealt with sequentially. This leads to
suboptimal solutions, especially in certain types of costs. 

We developed and implemented a BPP model for a Chilean flower producer, and
compared the results with the ones generated using a manual procedure. The
BPP model was solved as a mixed integer programming problem in about two
hours on a regular PC. The utilization of operation research methods to
solve agribusiness problems is growing, but it's still behind the
utilization in other industries. The executives of the participating company
adopted the BPP model and  committed to using it in the future. The experience gained with this company should help implement
similar system in other agribusinesses. 
\marginpar{Es muy chivero?}


\section{Acknowlegments}

The authors acknowledge the comments by William Foster and Rob A.C.M. Broekmeulen, and the support of the Pacific Flowers{\ company}. Finally, we
also acknowledge the support by the Comisi\'{o}n Nacional de Investigaci\'{o}n Cient\'{\i}fica y Tecnol\'{o}gica de Chile (CONICYT).

\bibliography{flowers_bib}
\bibliographystyle{apalike}
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